The early universe (about 800 million years after the big bang) contained atomic hydrogen gas. accretion of gas onto early uv light which ionized the surrounding hydrogen. consider a quasar uv source emitting n photons per second into hydrogen gas of number density n per cm cubed. each photon ionizes i hydrogen atom. (assume n and n to be constant in time) super massive black holes (quasars) emitted the quasar after time r? b) using this expression calculate the rate of expansion dr/dt. under what conditions dr/dt be larger than c? given the constant speed of light this expression must be incorrect. in fact to account for th finite speed of light we calculate the volume ionized as the difference between the volur contained within a radius equal to the light horizon at time t, and the volume ionized duri the light crossing time of the region (equal to r/c) at time t. the latter are subtracted beca those photons did not encounter any hydrogen atoms yet. c) derive the correct expression for dr/dt d) show that this expression always has dr'dt